Pairs of additive equations IV. Sextic equations
Pairs of additive equations of small degree
Pairs of additive forms and Artin's conjecture
Pairs of cubic forms in many variables
Pairs of Pell equations having at most one common solution in positive integers.
Palindromic powers.
Parametric solutions for some Diophantine equations.
Parametric Solutions of the Diophantine Equation A² + nB⁴ = C³
The Diophantine equation A² + nB⁴ = C³ has infinitely many integral solutions A, B, C for any fixed integer n. The case n = 0 is trivial. By using a new polynomial identity we generate these solutions, and then give conditions when the solutions are pairwise co-prime.
Parametrization of integral values of polynomials
We will recall a recent result about the classification of those polynomial in one variable with rational coefficients whose image over the integer is equal to the image of an integer coefficients polynomial in possibly many variables. These set is polynomially generated over the integers by a family of polynomials whose denominator is and they have a symmetry with respect to a particular axis.We will also give a description of the linear factors of the bivariate separated polynomial over a...
Parametrized solutions of Diophantine equations
Parametrizing SL₂(ℤ) and a question of Skolem
Partitions and indefinite quadratic forms.
Pell and Pell-Lucas numbers of the form
In this paper, we find all Pell and Pell-Lucas numbers written in the form , in nonnegative integers , , , with .
Perfect powers expressible as sums of two fifth or seventh powers
We show that the generalized Fermat equations with signatures (5,5,7), (5,5,19), and (7,7,5) (and unit coefficients) have no non-trivial primitive integer solutions. Assuming GRH, we also prove the non-existence of non-trivial primitive integer solutions for the signatures (5,5,11), (5,5,13), and (7,7,11). The main ingredients for obtaining our results are descent techniques, the method of Chabauty-Coleman, and the modular approach to Diophantine equations.
Perfect powers in arithmetic progression. A note on the inhomogeneous case
Perfect powers in arithmetical progression (II)
Perfect powers in linear recurring sequences
Perfect powers in products of integers from a block of consecutive integers
Perfect powers in products of integers from a block of consecutive integers (II)
Perfect powers in products of terms in an arithmetical progression III